Answer :

a, b, c, d are in G.P.


Therefore,


bc = ad … (1)


b2 = ac … (2)


c2 = bd … (3)


To prove: (a2 + b2), (b2 + c2), (c2 + d2) are in G.P, we need to prove that:


(a2 + b2) (c2 + d2) = (b2 + c2)2 {deduced using GM relation}


RHS = (b2 + c2)2 = b4 + c4 + 2b2c2


= a2c2 + b2d2 + a2d2 + b2c2 {using equation 2 and 3}


= c2(a2 + b2) + d2(a2 + b2)


= (a2 + b2) (c2 + d2) = LHS


(a2 + b2), (b2 + c2), (c2 + d2) are in G.P


Hence proved.


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